Talk:Lagrange point

L1, L2, & L3 Quintic Equations

The correct polynomial for the distance between the secondary and L1 is

x^5+(µ-3)x^4+(3-2µ)x^3-(µ)x^2+(2µ)x-µ = 0

and

r = R x

The correct polynomial for the distance between the secondary and L2 is

x^5+(3-µ)x^4+(3-2µ)x^3-(µ)x^2-(2µ)x-µ = 0

— Preceding unsigned comment added by 131.176.243.11 (talk • contribs) 09:03, 19 July 2023 (UTC)

I have verified that both our equations are correct with the MATLAB symbolic toolbox. Here's the details:

r=Rx

Using µ=m2/(m1+m2) where m1 is the mass of the Sun and m2 is the mass of the Earth for the existing force balance equations.

L1: x^5+(µ-3)x^4+(3-2µ)x^3-(µ)x^2+(2µ)x-µ = 0

L2: x^5+(3-µ)x^4+(3-2µ)x^3-(µ)x^2-(2µ)x-µ = 0

As you correctly specified.

Using µ=m1/(m1+m2) (I previously called this expression µ because that's how the existing Wikipedia page was at the time) where m1 is still the mass of the Sun and m2 is the mass of the Earth for the existing force balance equations.

L1: x^5-(2+µ)x^4+(1+2µ)x^3+(µ-1)x^2-2(µ-1)x+µ-1

L2: x^5+(2+µ)x^4+(1+2µ)x^3+(µ-1)x^2+2(µ-1)x+µ-1

As I correctly specified. WaffleJet34 (talk) 18:00, 11 July 2024 (UTC)

L1 Quintic Equation:

This equation is incorrect. To demonstrate, take the arbitrary test case where R = 6; M1 = 10, and M2 = 2, (µ = .8333) the quintic equation has a zero at x=0.8098. Since x = r/R, then r = 6*0.8098 = 4.8586. Plugging these values into the formula above the quintic equation describing the force balance, the left-hand-side becomes 10/(6-4.8586)^2-2/4.8586^2 = 7.5911. The right-hand-side equation becomes (10/(10+2)*6-4.8586)*(10+2)/6^3=0.0079. Clearly these aren't equal so the quintic equation is incorrect.

Using MATLAB's symbolic toolbox, the correct equation was derived to be: x^5-(2+µ)x^4+(1+2µ)x^3+(µ-1)x^2-2(µ-1)x+µ-1. The only difference is the minus sign in front of the coefficient of the x-term. Taking again the previous test case, this equation has a zero at x=0.3414. Since x = r/R, then r = 6*0.3414 = 2.0487. Plugging these values into the formula above the quintic equation describing the force balance, the left-hand-side becomes 10/(6-2.0487)^2-2/2.0487^2 = 0.1640. The right-hand-side equation becomes (10/(10+2)*6-2.0487)*(10+2)/6^3=0.1640. These are equal, verifying the validity of the proposed equation.

L2 Quintic Equation:

Again, this equation is incorrect. First of all, instead of x, it has r as the variable. This results in mixed units throughout the equation as r is not dimensionless. Assuming the r's were supposed to be x's it is still incorrect, however. Taking the same test case as before, the quintic equation has a root at x=0.9025. Since x = r/R, then r = 6*0.9025 = 5.4150. Plugging these values into the formula above the quintic equation describing the force balance, the left-hand-side becomes 10/(6+5.4150)^2+2/5.4150^2 = 0.1450. The right-hand-side equation becomes (10/(10+2)*6+5.4150)*(10+2)/6^3=0.5786. Clearly these aren't equal so the quintic equation is incorrect.

Using MATLAB's symbolic toolbox, the correct equation was derived to be: x^5+(2+µ)x^4+(1+2µ)x^3+(µ-1)x^2+2(µ-1)x+µ-1. This is very similar to the quintic equitation for L1 except all the coefficients are positive. The only difference is the minus sign in front of the coefficient of the x-term. Taking again the previous example, this equation has a zero at x=0.4381. Since x = r/R, then r = 6*0.4381 = 2.6285. Plugging these values into the formula above the quintic equation describing the force balance, the left-hand-side becomes 10/(6+2.6285)^2+2/2.6285^2 = 0.4238. The right-hand-side equation becomes (10/(10+2)*6+2.6285)*(10+2)/6^3=0.4238. These are equal, verifying the validity of the proposed equation.

L3 Quintic Equation

The equation was not written. Once again using the MATLAB symbolic toolbox, the resulting quintic equation is x^5+(µ-8)x^4+(25-6µ)x^3+(13µ-37)x^2+2(13-7µ)x+7(µ-1). Taking again the previous example, this equation has a zero at x=0.0975. Since x = r/R, then r = 6*0.0975 = 0.5850. Plugging these values into the formula describing the force balance in the L3 section, the left-hand-side becomes 10/(6- 0.5850)^2+2/(2*6- 0.5850)^2=0.3564. The right-hand-side equation becomes (2/(10+2)*6+6-0.5850)*(10+2)/6^3=0.3564. These are equal, verifying the validity of the proposed equation. WaffleJet34 (talk) 03:57, 24 May 2023 (UTC)

Equation rendering not working

Some equations only appear as LaTeX script. Not being an expert, I cannot see what it missing to permit proper rendering. Ccison (talk) 13:14, 16 August 2024 (UTC)

The equations look fine on my computer. Can you be specific about which ones are not rendering for you? Tercer (talk) 18:48, 16 August 2024 (UTC)

Apologies for responding so late to your question. It now appears that the rendering problem was on my end. After some updates, this problem seems to no longer be a problem. Ccison (talk) 22:33, 7 May 2025 (UTC)

Date formats

@Tercer: Okay, so we're obviously at an impasse on the date formats. I guess I'll be the one to start a proper dialog. We're not seeing eye-to-eye and I don't really want to write an essay explaining what happened yesterday or why. Suffice to say Tercer's original edit summary here was 100% unhelpful, ownership-like behavior and really ticked me off. Always give a valid reason when reverting anything other than vandalism; that's WP:REVEXP. Also, exactly what part of WP:CAUGHTUP don't you understand? I cited that suggestion here and it was again ignored afterwards. I don't get it. May I at least replay my other edits not related to date formats?

Anyway, to the point of date formats, your position is that I shouldn't change the established date format without getting a consensus first on the talk page. I get that and fully agree with that point, and you are correct that it is stated in MOS:DATERET right where it says, "Unless there are reasons for changing it based on the topic's strong ties to a particular English-speaking country, or consensus on the article's talk page."

What I've pointed out in my edit summaries, and for some reason you're still not getting, is that this article does not consistently apply one date format; it has three. I took the predominant one, as I've done 1000 times, and applied it consistently throughout the article as DATERET says to do in basically all three of its bullet points. So I'm not sure why the resistance to consistency. Is it that you don't like MDY? Because at this point I'll happily go with DMY. But let's please choose a date format and apply it consistently throughout the article. This is too trivial to waste much more time on. — voidxor 14:14, 7 May 2025 (UTC)

Fair enough, I have partially restored your WP:CAUGHTUP edit. I didn't think it was worth preserving, since it was mostly automated non-visible changes. The substantial part was deleting a true but unsourced statement. Instead of eliminating it I added a source.

As for the date format, the problem is that there isn't a predominant one. If anything yyyy-mm-dd is the most commonly used, and that's what you should use if you absolutely need to standardize it. But I really don't think you should standardize it.

I do apologize for my initial unhelpful edit summary. I got angry because I really don't want to waste any time arguing about date formats, and I just wanted it to go away. Of course, the effect was the exact opposite. Tercer (talk) 16:05, 7 May 2025 (UTC)

Thank you for your comments above; they make me feel a lot better! Thanks as well for finding and adding a reference! My only other thoughts are twofold:

• You keep referring to DATERET, which emphasizes consistency. MOS:DATEUNIFY goes into more detail. If fixing it on this article is going to cause stress right now, it can wait.
• yyyy-mm-dd, which comes from the ISO 8601 standard, is not acceptable in prose. Per MOS:DATE, it can only be used "in limited situations where brevity is helpful". That includes tables and maybe citations, though I would argue DATEUNIFY should prevail and we should use the same format in prose and citations. That's why articles can be tagged as using DMY or MDY dates but not YMD. Likewise, the script I'm using doesn't support YMD. — voidxor 16:29, 7 May 2025 (UTC)

I'm glad we could have come to a civil understanding. If the issue is prose versus citations, though, the solution is simple: there is a single date in the entire article, and it is in the DMY format. It is therefore trivially consistent, and there's nothing to fix.

MOS:DATEUNIFY does not prescribe that the dates in prose and the citations must have the same format; it is explicit that they might be different. I think you have three options then (1) Leave them as they are, (2) Standardize to the predominant style yyyy-mm-dd, (3) Change to the prose style DMY. Tercer (talk) 16:51, 7 May 2025 (UTC)

Okay. Yesterday I'd counted overall; I didn't break it out by prose versus citations. YMD should be avoided in prose. With the script I can easily switch the whole article to DMY, if that's cool with you. — voidxor 17:31, 7 May 2025 (UTC)

I won't revert you, if that's what you're asking. Tercer (talk) 17:49, 7 May 2025 (UTC)

 Done, thanks. — voidxor 19:10, 7 May 2025 (UTC)

Clarify what the "Mass Ratio" is

I have seen the "mass ratio" defined two separate and opposite ways elsewhere and there is a further link on a page sidebar to an entire separate "mass ratio" that is about rockets. It would he helpful if there was a short formula there or something Collapsedsquid (talk) 09:07, 6 November 2025 (UTC)

It's nothing deep, it's just the ratio between the masses. Tercer (talk) 11:25, 6 November 2025 (UTC)

Is bigger over smaller or smaller over bigger? if you get it wrong than you have exactly inverted the result. I also see it represented elsewhere as M2/(M1+M2) which for these purposes is identical but is not "ratio between the two masses."

I should also say that I had this issue in the context of discussion on the effect of the mass ratio on the instability level of L1/L2/L3 points which I realize may be complicated to get into in the article lead before you go into equations but maybe a short point about how the mass ratio affects that would be helpful. Collapsedsquid (talk) 19:07, 6 November 2025 (UTC)

External link proposal: interactive educational visualization of Lagrange points

Hello!

For transparency, I’m the author of https://lagrange-points.thesmirnov.com and I’d appreciate an independent review of whether this link is appropriate for the “External links” section.

Proposed link:

• Interactive visualization of Lagrange points — rotating-frame two-body system; adjustable mass ratio and vector visualization of gravitational and centrifugal components

Why it may be useful to readers:

1. Directly relevant to the article topic (L1–L5 in a rotating two-body system).
2. It is an educational interactive visualization (not a source for article facts).
3. Readers can adjust the mass ratio and inspect the component vectors and their resultant.
4. This is a proposal for one link only.

I’d appreciate an independent assessment and I’m happy to adjust the wording if needed. TheSmirnov (talk) 12:54, 24 February 2026 (UTC)